A hypergeometric function transform and matrix-valued orthogonal polynomials

TL;DR

This paper develops a spectral decomposition for a specific differential operator, leading to a generalized Fourier transform with hypergeometric functions and introducing matrix-valued orthogonal polynomials as generalizations of Wilson polynomials.

Contribution

It provides an explicit spectral analysis of a second-order differential operator and constructs matrix-valued orthogonal polynomials related to Wilson polynomials.

Findings

Spectral decomposition includes continuous and discrete parts with specified multiplicities.

Derived a generalized Fourier transform with hypergeometric kernel.

Established orthogonality relations for matrix-valued polynomials.

Abstract

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with multiplicity one. The spectral analysis gives rise to a generalized Fourier transform with an explicit hypergeometric function as a kernel. Using Jacobi polynomials the operator $T$ can also be realized as a five-diagonal operator, hence leading to orthogonality relations for $2\times 2$-matrix-valued polynomials. These matrix-valued polynomials can be considered as matrix-valued generalizations of Wilson polynomials.